I removed the flagstick and placed it on the ground behind the hole. After I putted my fellow competitor thought that my ball might hit the flagstick and moved it whilst my ball was moving. Does he incur a penalty?
He would incur a penalty of 2 stokes (or loss of hole in Matchplay) for removing an obstruction which might influence the movement of the ball. (Decision of the Rules of Golf 17-1/7)
This rule seems harsh as he was only trying to help you but unfortunately the Rules of Golf are very clear on this point.
If, however, your fellow competitor was attending the flagstick removed it from the hole, placed it on the ground and then moved it there is no penalty. (Decision of the Rules of Golf 17-1/6)